19^ On the RESOLUTION of 



pofing ni'y+y — iti'b — zvia — b, that is, y — 



m^h — ima — b 



— ^jjl^; — . But X •=. mb — my — a, and fubftituting, x r: 

 — m'^+i — • Thus, if rt = 5, and ^ z: lo, and m zz 2 ; then 



4.10— 4.5— 10 4.5+4.10— j / \ , 



J = J- =z 2,, and X = -^-^^— ^ =z 11; but (11 )"-+ 



(2)^ = 125 = {loY+isr- 



Cor. If b = o, we fhall obtain two fquares, the fum of which 

 fliall be a given fquare. For j =z — m'+i ' °^ ~^ m^'+i ' ^^^ * — 



4.10 



t^j ♦ Thus, if rt z: 10, and «z = 2, then j/ r: — - — =■ 8, and 



— ^— - — =z 6, but 64+36 = 100. 



PROBLEM III. 



To find two rational numbers, the fquares of which, together with 

 any given multiple of their product, fhall be equal to a given fquare. 



By hypothefis, x'^-{-y''-\-bxy z= a^, and tranfpofing x''-\-bxy zz 

 a^ — y'^, and refolving into fadlors, x{x-\-by^ zz [a-^y){a — -v) ; 



whence, by afTumption, x-\-by zz ma — my, and x ■=. . 



Tranfpofing the firfl: equation, x n ma — my — by j confequent- 



ly, — '— zz ma — my — by, or a-\-y zz m'^a — my — mby, and again 



by tranfpofing, m'^y-\-mby-\-y zz m'^a — a ; whence y zi 



m* — I fl+v , . 2m+b 



— .. L, -Xa. But X =: — ^whererore x =. — > , , . Xa. 



m +mb+l m ' m^+mb+1 



Suppose 



